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# club

If $\kappa$ is a cardinal then a set $C\subseteq\kappa$ is *closed* iff for any $S\subseteq C$ and $\alpha<\kappa$, $\sup(S\cap\alpha)=\alpha$ then $\alpha\in C$. (That is, if the limit of some sequence in $C$ is less than $\kappa$ then the limit is also in $C$.)

If $\kappa$ is a cardinal and $C\subseteq\kappa$ then $C$ is *unbounded* if, for any $\alpha<\kappa$, there is some $\beta\in C$ such that $\alpha<\beta$.

If a set is both closed and unbounded then it is a *club* set.

Defines:

club, closed, unbounded, closed unbounded, closed set, unbounded set, closed unbounded set, club set

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03E10*no label found*

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## Attached Articles

club filter by Henry

## Corrections

linking policy by asteroid ✓